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Register a new userThe central configuration of the planar ($N$+1)-body problem with a regular $N$-polygon
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For planar ($N$+1)-body ($N$,$geq$ 2) problem with a regular $N$-polygon, under the assumption that the ($N$+1)-th body locates at the geometric center of the regular $N$-polygon, we obtain the sufficient and necessary conditions that the $N$+1 bodies can form a central configuration.
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We introduce an algebraic method to study local stability in the Newtonian $n$-body problem when certain symmetries are present. We use representation theory of groups to simplify the calculations of certain eigenvalue problems. The method should be applicable in many cases, we give two main examples here: the square central configurations with four equal masses, and the equilateral triangular configurations with three equal masses plus an additional mass of arbitrary size at the center. We explicitly found the eigenvalues of certain 8x8 Hessians in these examples, with only some simple calculations of traces. We also studied the local stability properties of corresponding relative equilibria in the four-body problems.
We study the dynamics of a class of endomorphisms of A^N which restricts, when N = 1, to the class of unicritical polynomials. Over the complex numbers, we obtain lower bounds on the sum of Lyapunov exponents, and a statement which generalizes the compactness of the Mandelbrot set. Over the algebraic numbers, we obtain estimates on the critical height, and over general algebraically closed fields we obtain some rigidity results for post-critically finite morphisms of this form.
For the Newtonian (gravitational) $n$-body problem in the Euclidean $d$-dimensional space, the simplest possible solutions are provided by those rigid motions (homographic solutions) in which each body moves along a Keplerian orbit and the configuration of the $n$-body is a constant up to rotations and scalings named textit{central configuration}. For $dleq 3$, the only possible homographic motions are those given by central configurations. For $d geq 4$ instead, new possibilities arise due to the higher complexity of the orthogonal group $O(d)$, as observed by Albouy and Chenciner. For instance, in $mathbb R^4$ it is possible to rotate in two mutually orthogonal planes with different angular velocities. This produces a new balance between gravitational forces and centrifugal forces providing new periodic and quasi-periodic motions. So, for $dgeq 4$ there is a wider class of $S$-textit{balanced configurations} (containing the central ones) providing simple solutions of the $n$-body problem, which can be characterized as well through critical point theory. In this paper, we first provide a lower bound on the number of balanced (non-central) configurations in $mathbb R^d$, for arbitrary $dgeq 4$, and establish a version of the $45^circ$-theorem for balanced configurations, thus answering some questions raised by Moeckel. Also, a careful study of the asymptotics of the coefficients of the Poincare polynomial of the collision free configuration sphere will enable us to derive some rather unexpected qualitative consequences on the count of $S$-balanced configurations. In the last part of the paper, we focus on the case $d=4$ and provide a lower bound on the number of periodic and quasi-periodic motions of the gravitational $n$-body problem which improves a previous celebrated result of McCord.
In this paper, we show that there is a Cantor set of initial conditions in the planar four-body problem such that all four bodies escape to infinity in a finite time, avoiding collisions. This proves the Painlev{e} conjecture for the four-body case, and thus settles the last open case of the conjecture.
In this paper, we first describe how we can arrange any bodies on Figure-Eight without collision in a dense subset of $[0,T]$ after showing that the self-intersections of Figure-Eight will not happen in this subset. Then it is reasonable for us to consider the existence of generalized solutions and non-collision solutions with Mixed-symmetries or with Double-Eight constraints, arising from Figure-Eight, for N-body problem. All of the orbits we found numerically in Section ref{se7} have not been obtained by other authors as far as we know. To prove the existence of these new periodic solutions, the variational approach and critical point theory are applied to the classical N-body equations. And along the line used in this paper, one can construct other symmetric constraints on N-body problems and prove the existence of periodic solutions for them.
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